11.12.2009 - BIRS: Noise, Time Delay and Balance Control

Tamas Insperger

System under consideration
- \dot{x} = Ax + Bu
- y = Cx
  - y is only known by y(t-\tau)
- u = f(r-y)
  - u = Dy(t-\tau)
- \ddot{x}(t) +a_1\dot{x}(t)+a_0x(t)=u(t)
  - linear approximation u(t)=px(t-\tau)+p\tau\dot{x}(t-\tau)
  - secant approximation u(t)=2px(t-\tau)+p\tau x(t-2\tau)
  - integral approximation u(t)=\int_0^{t-\tau} w_0 x(s) + w_1\dot{x}(s) ds
Smith predictor
- x = \frac{1}{s-a}u
- s-a+b e^{\tau s} or \dot{x}(t) + a x(t) - b x(t-\tau)
- becomes \ddot{x}(t) +(b-2a)\dot{x}(t) + a(a-b)x(t) if estimated delay and state are same as the real delay and state
- Smith predictor is sensitive to parameters and not necessarily good to stabilize unstable systems
Predictive control
- Use the available delayed state to predict the current state
Delayed feedback
- Use the delayed output directly
Classification of linear systems
- Time invariant ODE
  - n dimensions = n eigen-values
- Time invariant DDE
  - n dimensions = infinite number of eigen-values
- Time periodic ODE
  - On one period (need to know the map) then the there are n eigen-values for n-dimensions of the period
  - \dot{x}(t)=A(t)x(t)
  - A(t+T) = A(t)
  - x(T) = \Phi x(0)
- Time periodic DDE
  - Infinite number of eigen-values
Brockett problem
- \dot{x}(t) = A x(t) + BG(t)Cx(t-\tau)
Act and wait
- G(t) = 0 during waiting period, which is longer than delay
- G(t) = gg(t) during acting period, which is shorter than delay
- Step-by-step solution
  - \dot{x}(t)=Ax(t)
  - x(t)=\Phi^1(t)x(0)
  - \Phi^1(t)=e^{At}
  - \Phi^2(t)=e^{At}+\int_{tw}^t e^{A(t-s)} B \Gamma(s) C e^{A(s-/tau)} ds
  - \Phi^3(t)=\Phi^2(t)+stuff
  - x(T)=\Phi^3(T)x(0)
    - This ends up with in n-poles.

Francisco Valero-Cuevas

Internal models
Predictive strategies
State estimators
Development in childhood
Biological computation

James Finley and Eric Perreault

Feedforward vs feedback control during balance
Heightened co-contraction (Hogan 1984; Milner 2002)
Larger gain on stretch reflexes during a "compliant" environment
Co-contraction strategy
- see co-contraction increase in "unstable" condition
- co-contraction appears to inhibit stretch reflex
- Ankle stiffness increases as stability of joint decreases
- What about signs on net stiffness? -Kank-Kenv+mgl
There appears to be a bias towards feedforward over feedback control

Tim Kiemel and John Jeka

Linearized "unstable" pendulum model with delayed PD control
- Plant is the mapping from EMG to body segment angles
- Feedback is the mapping between changes in body angles into EMG
- Intrinsic stiffness vs ankle stiffness
  - stability achieved with a combination of hip and ankle stiffness
- Feedback in the nervous system is probably not PD, most likely something better

Meeting thoughts

Intermittent vs Continuous and Linear vs Non-linear
- How do these fundamental questions help/hurt modeling of posture?
Kleinman DL, "Optimal control with time delay and observation noise." IEEE Trans Automatic Control (15)524-527, 1969
Palmor ZJ (1996) Time delay compensation smith predictor and its modifications. Levine "The control handbook", CRC Press

Gabor Stepan

Chaos is amusing. This means that there needs to be large non-linearities
Digital control systems introduce "spatial" and "temporal" delays.
There is a point where there is "micro-chaos" due digitization where the system is caught in an oscillation before it gets to the regulated point.
Are the delays in the human system a continuous delay or a discrete delay?
Delayed feedback on jerk!

Questions for future math problems

Instead of linearizing equations that have noise in them transfer the system into probability space where the equations are linear in probability
- Delayed Langevin equation
Try and determine if you can tell if a system is linear or non-linear from time-series data